Area and Perimeter Calculator

What this calculator answers

How much surface a flat shape covers (its area) and how far it is around the edge (its perimeter), worked out together for rectangles, squares, parallelograms, trapezoids and ellipses. The area tells you how much flooring, sod or fabric covers the shape; the perimeter tells you how much baseboard, edging or fence goes around it. Triangles and circles have their own calculators.

How to use it

• Shape: pick the shape. The drawing in the results changes with it.
• Length unit: one unit for every measurement. In feet you can type decimal feet (12.5) or feet and inches (12' 6", 12 ft 6 in, 12 6); the line under each box says how it was read. Switching units keeps the numbers you typed, so pick the unit your tape measure shows.
• Decimal places: how the results are rounded for display. The calculation keeps full precision.
• Given (parallelogram and trapezoid): what you measured. A parallelogram needs its base and height for the area, and its slanted side for the perimeter. A trapezoid can be described by its bases and height (with equal legs or one right-angle leg), by its bases, height and both legs, or by all four sides.
• Lengths: each box is labeled on the drawing. For an ellipse, a and b are half the overall length and width, measured from the center.

Results update as you type. The Try buttons load the other cases worked through below: a 12 × 12 ft room, an oval table, a leaning parallelogram and a trapezoid measured on its four sides. Save for comparison keeps up to three shapes side by side.

Area and perimeter formulas for common shapes

• $L$ and $W$ are the length and width; $s$ is a side.
• $b$ is a parallelogram’s base, $s$ its slanted side and $h$ its height, measured at right angles to the base.
• $a$ and $b$ are a trapezoid’s parallel sides (the bases), $c$ and $d$ its legs and $h$ the distance between the bases.
• For an ellipse, $a$ and $b$ are the semi-axes: half the overall length and half the overall width.

Worked example: a 12 ft 4 in × 10 ft 6 in bedroom

1. Convert the inches to feet: 12 ft 4 in = 12 + 4 ÷ 12 = 12.3333 ft, and 10 ft 6 in = 10.5 ft.
2. Area: A = 12.3333 × 10.5 = 129.5 ft².
3. Perimeter: P = 2 × (12.3333 + 10.5) = 45.67 ft, which is 45 ft 8 in.
4. Diagonal, corner to corner: √(12.3333² + 10.5²) = 16.2 ft.
5. In square meters: 129.5 × 0.09290304 = 12.03 m².

A square room with the same 45 ft 8 in perimeter would have 11 ft 5 in sides and 130.34 ft² of floor, only 0.84 ft² more: this bedroom is already 99.4% of the most floor that length of wall can enclose.

Area and perimeter of a rectangle or square

Multiply the length by the width for the area; add the four sides, 2 × (length + width), for the perimeter. A square is the case where both are the same, so a 12 ft square has 144 ft² and a 48 ft perimeter.

The same perimeter can hold very different areas. The Rectangles with the same perimeter table keeps your perimeter and changes the proportions: the square always encloses the most, and a strip nine times as long as it is wide encloses only 36% of the square’s area. That matters when a fixed length of fence or edging has to surround as much ground as possible.

To check that a real room is rectangular, measure both diagonals. They match only when the corners are square; if they differ, the area of the rectangle is an estimate.

Area and perimeter of a parallelogram

The area is the base times the height, where the height is measured at right angles to the base, not along the slanted side. The perimeter is 2 × (base + side). Cut the slanted end off one side and slide it to the other, and the parallelogram becomes a rectangle with the same base and height, so the area follows.

For a parallelogram with an 8 cm base and 5 cm sides at 60°, the height is 5 × sin 60° = 4.33 cm, the area is 8 × 4.33 = 34.64 cm² and the perimeter is 2 × (8 + 5) = 26 cm. Its diagonals are 7 cm and 11.36 cm. Multiplying the base by the side instead gives 40 cm², 15.5% too much: that is the area only when the angle is 90°. The same sides at other angles table shows the area falling as the shape leans further (20 cm² at 30°) while the perimeter stays at 26 cm.

The height can never be longer than the side, because the side slants. With a 10 ft base, a 5 ft side and a 4 ft height, the angles are 53.13° and 126.87°; a height over 5 ft is refused. If you know only the base and height, choose Base and height only: you get the area, and the lowest the perimeter could be, 2 × (base + height).

How to find the perimeter of a trapezoid (with or without the legs)

The area needs only the two bases and the height: A = (a + b) ÷ 2 × h. The perimeter also needs the two legs, and the calculator can work them out when the trapezoid is a common kind:

• Equal legs (isosceles): each leg runs half the difference of the bases along the bottom, so leg = √(h² + ((a − b) ÷ 2)²). With bases of 14 ft and 8 ft and a 4 ft height, each leg is √(4² + 3²) = 5 ft, the area is 44 ft² and the perimeter is 14 + 8 + 5 + 5 = 32 ft.
• One right-angle leg: that leg equals the height, and the other is √(h² + (a − b)²). The same bases and height give legs of 4 ft and 7.21 ft and a 33.21 ft perimeter.
• All four sides: the height can come from the sides alone. Slide one leg across the top base and it forms a triangle with the other leg and the difference between the bases; that triangle’s height is the trapezoid’s. A trapezoid with bases of 12 m and 7 m and legs of 5 m and 6 m is 4.8 m high, so its area is (12 + 7) ÷ 2 × 4.8 = 45.6 m² and its perimeter 30 m.

A leg can never be shorter than the height, because the height is the shortest distance between the bases; the calculator says so instead of giving an answer. When you enter the bases, the height and both legs, it also checks that the five numbers describe one shape. Legs of 5 ft and 6 ft don’t fit the 14 ft and 8 ft bases at a 4 ft height: those four sides would need a height of about 4.55 ft, so the result is withheld until the measurements agree within 1%.

Perimeter of an ellipse: why there’s no simple formula

The area of an ellipse is simply π × a × b, but its perimeter is an elliptic integral, which has no formula built from a few ordinary operations. This calculator computes it with Gauss’s arithmetic-geometric mean, a quickly converging series that is exact to the precision shown (about 15 significant digits), and then compares the common approximations with it.

For an oval table 60 in × 36 in, the semi-axes are 30 in and 18 in, so the area is 540π = 1,696.46 in² (11.78 ft²) and the perimeter is 153.16 in, about 12 ft 9 in of edge banding.

Ramanujan’s second formula is the one the results show next to the exact value: it always comes out slightly low, and for everyday shapes the gap is far too small to measure. The two shortcuts are off by about 1.5% here, and by more for longer, narrower ellipses.

Area vs. perimeter: units and common mix-ups

Perimeter is a length, so it is in feet, meters or another length unit. Area is in square units: a square foot is a 1 ft × 1 ft square. Converting an area needs the square of the length factor. Since 1 ft = 0.3048 m, 1 ft² = 0.3048² = 0.09290304 m², so the 129.5 ft² bedroom is 12.03 m², not 129.5 × 0.3048 = 39.47. In the same way 1 yd² = 9 ft², and an acre is 43,560 ft².

The Area and perimeter in other units table converts your result into every unit on the page, including acres and hectares, with exact factors.

Reading the result

• Area is the headline, in the square of your unit, with the shape restated underneath.
• Perimeter comes first under it, also in feet and inches when you work in feet or inches.
• Area in another unit is the area in the most useful second unit (square meters for feet, square feet for inches or meters), with a third unit under it and acres or hectares for large areas.
• The third box depends on the shape: the diagonal of a rectangle, the angles or height of a parallelogram, the legs of a trapezoid, or Ramanujan’s approximation for an ellipse with its gap from the exact perimeter.
• The drawing is to scale with your measurements labeled, unless one side is more than six times the other; it then says it is not to scale.
• How this was calculated puts your measurements into each formula, starting with any feet-and-inches conversion.
• The shape’s table: rectangles with the same perimeter, a parallelogram at other angles, or the ellipse perimeter formulas compared. Each downloads as a CSV file.
• Continue in the Flooring and Tile Calculator (shown for inches, feet, yards and metric units up to meters) carries the area over as a total floor area, where the waste allowance and boxes are added.

Assumptions and limitations

• Every length is in the one unit you choose; mixed units such as meters and centimeters need converting first.
• Sides are straight and the shape is flat. Rounded corners, wall thickness and trim are not counted.
• A rectangle’s corners are right angles, a parallelogram’s opposite sides are parallel and equal, and a trapezoid’s bases are parallel.
• Triangles, circles, sectors and regular polygons are not included; use a triangle or circle calculator for those.
• Results are rounded for display only.

Common mistakes

• Typing 12.6 for 12 ft 6 in. 12.6 ft is 12 ft 7.2 in. Type 12 6, 12' 6" or 12.5; the line under the box shows how it was read.
• Entering the full width of an ellipse as a. a and b are half the overall size. A 60 in oval has a = 30, not 60; the headline restates the overall size so a doubled entry stands out.
• Using the slanted side as a parallelogram’s height. That gives base × side, which is too large whenever the shape leans.
• Assuming a trapezoid’s legs are equal. The equal-legs option is a common textbook case; measure the legs if the sides aren’t symmetrical.
• Converting square units with a length factor. Square feet to square meters is × 0.09290304, not × 0.3048.